Difference between revisions of "AMC 12C 2020 Problems"

Latest revision as of 14:02, 27 November 2022

1 plus 1

==Problem 1

A plane flies at a speed of $590$ miles/hour. How many miles in two hours

Problem 3

In a bag are $7$ marbles consisting of $3$ blue marbles and $4$ red marbles. If each marble is pulled out $1$ at a time, what is the probability that the $6th$ marble pulled out red?

$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ \frac{1}{8} \qquad\textbf{(C)}\ \frac{1}{2} \qquad\textbf{(D)}\ \frac{4}{7} \qquad\textbf{(E)}\ 1$

Problem 4

A spaceship flies in space at a speed of $s$ miles/hour and the spaceship is paid $d$ dollars for each $100$ miles traveled. It’s only expense is fuel in which it pays $\frac{d}{2}$ dollars per gallon, while going at a rate of $h$ hours per gallon. Traveling $3s$ miles, how much money would the spaceship have gained?

$\textbf{(A)}\ 20 \qquad\textbf{(B)}\ \ 21 \qquad\textbf{(C)}\ \ 22 \qquad\textbf{(D)}\ \ 23 \qquad\textbf{(E)}\ 24$

Problem 5

Let $R(x)$ be a function satisfying $R(m + n) = R(m)R(n)$ for all real numbers $n$ and $m$ . Let $R(1) = \frac{1}{2}.$ What is $R(1) + R(2) + R(3) + … + R(1000)$ ?

Problem 6

How many increasing(lower to higher numbered) subsets of $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ contain no $2$ consecutive prime numbers?

Problem 7

A Regular Octagon has an area of $18 + 18\sqrt {2}$ . What is the sum of the lengths of the diagonals of the octagon?

Problem 8

What is the value of $sin(1^\circ)sin(3^\circ)sin(5^\circ)…sin(179^\circ) - sin(181^\circ)sin(182^\circ)…sin(359^\circ)$ ?

Problem 9

Let $E(x)$ denote the sum of the even digits of a positive integer and let $O(x)$ denote the sum of the odd digits of a positive integer. For some positive integer $N$ , $3E(3N)$ = $4O(4N)$ . What is the product of the digits of the least possible such $N$ ?

Problem 10

In how many ways can $n$ candy canes and $n + 1$ lollipops be split between $n - 4$ children if each child must receive atleast $1$ candy but no child receives both types?

Problem 11

Let $ABCD$ be an isosceles trapezoid with $\overline{AB}$ being parallel to $\overline{CD}$ and $\overline{AB} = 5$ , $\overline{CD} = 15$ , and $\angle ADC = 60^\circ$ . If $E$ is the intersection of $\overline{AC}$ and $\overline{BD}$ , and $\omega$ is the circumcenter of $\bigtriangleup ABC$ , what is the length of $\overline{E\omega}$ ? Source: JHMMC 2019

$\textbf{(A)} \frac {31}{12}\sqrt{2} \qquad \textbf{(B)} \frac {35}{12}\sqrt{3} \qquad \textbf{(C)} \frac {37}{12}\sqrt{5} \qquad \textbf{(D)} \frac {39}{12}\sqrt{7} \qquad \textbf{(E)} \frac {41}{12}\sqrt{11} \qquad$

Problem 12

For some positive integer $k$ , let $k$ satisfy the equation

$log(k - 2)! + log(k - 1)! + 2 = 2 log(k!)$ . What is the sum of the digits of $k$ ?

Problem 13

An alien walks horizontally on the real number line starting at the origin. On each move, the alien can walk $1$ or $2$ numbers the right or left of it. What is the expected distance from the alien to the origin after $10$ moves?

Problem 14

Let $K$ be the set of solutions to the equation $(x + i)^{10} = 1$ on the complex plane, where $i = \sqrt -1$ . $2$ points from $K$ are chosen, such that a circle $\Omega$ passes through both points. What is the least possible area of $\Omega$ ?

Problem 15

Let $N = 10^{10^{100…^{10000…(100 zeroes)}}}$ . What is the remainder when $N$ is divided by $629$ ?

Problem 16

Let $V$ and $F$ be the vertex and focus of the Parabola $P(x) = \frac{1}{8} x$ respectively. For a point $G$ lying on the directrix of $P(x)$ , and a point $H$ lying on $P(x)$ , $\overline {GH} = 10$ and Quadrilateral $VFGH$ is cyclic. If $VFGH$ has integral side lengths, what is the minimum possible area of $VFGH$ ?

Problem 17

Let $H(n)$ denote the $2nd$ nonzero digit from the right in the base - $10$ expansion of $(2n + 1)!$ , for example, $H(2) = 1$ . What is the sum of the digits of $\prod_{k = 1}^{2020}H(k)$ ?

Problem 18

$\bigtriangleup ABC$ lays flat on the ground and has side lengths $\overline{AB} = 3, \overline{BC} = 4$ , and $\overline{AC} = 5$ . Vertex $A$ is then lifted up creating an elevation angle with the triangle and the ground of $60^{\circ}$ . A wooden pole is dropped from $A$ perpendicular to the ground, making an altitude of a $3$ Dimensional figure. Ropes are connected from the foot of the pole, $D$ , to form $2$ other segments, $\overline{BD}$ and $\overline{CD}$ . What is the volume of $ABCD$ ?

$\textbf{(A) } 180\sqrt{3} \qquad \textbf{(B) } 15 + 180\sqrt{3} \qquad \textbf{(C) } 20 + 180\sqrt{5} \qquad \textbf{(D) } 28 + 180\sqrt{5} \qquad \textbf{(E) } 440\sqrt{2}$

Problem 19

Let $P(x)$ be a cubic polynomial with integral coefficients and roots $\cos \frac{\pi}{13}$ , $\cos \frac{5\pi}{13}$ , and $\cos \frac{7\pi}{13}$ . What is the least possible sum of the coefficients of $P(x)$ ?

Problem 20

What is the maximum value of $\sum_{k = 1}^{6}(2^{x} + 3^{x})$ as $x$ varies through all real numbers to the nearest integer?

$\textbf{(A)}\ -3\qquad\textbf{(B)}\ -2\qquad\textbf{(C)}\ -1\qquad\textbf{(D)}\ 0\qquad\textbf{(E)}\ 1$

Problem 21

Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$ . How many positive integers $x < 2020$ , satisfy the equation

Problem 22

A convex hexagon $ABCDEF$ is inscribed in a circle. $\overline {AB}$ $=$ $\overline {BC}$ $=$ $\overline {AD}$ $=$ $2$ . $\overline {DE}$ $=$ $\overline {CF}$ $=$ $\overline {EF}$ $=$ $4$ . The measure of $\overline {DC}$ can be written as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is $m + n$ ?

Problem 23

Problem 24

A sequence $(a_n)$ is defined as $a_1 = \frac{1}{\sqrt{3}}$ , $a_2 = \sqrt{3}$ , and for all $n > 1$ ,

$a_{n + 1} = \frac{2a_n-1}{1 - a_n^2}$

What is $\lfloor \ a_{2020}\rfloor$ ?

Problem 25

Let $P(x) = x^{2020} + 2x^{2019} + 3x^{2018} + … + 2019x^{2} + 2020x + 2021$ and let $Q(x) = x^{4} + 2x^{3} + 3x^{2} + 4x + 5$ . Let $U$ be the sum of the $kth$ power of the roots of $P(Q(x))$ . It is given that the least positive integer $y$ , such that $3^{y} > U$ is $2021$ . What is $k$ ?

@@ Line 1: / Line 1: @@
-==Problem 1==
+plus 1
-What is the the sum of the solutions of the equation <math>(x + 4)(x - 5)(x + 6)(x - 8) = 0</math>?
+==Problem 1
+A plane flies at a speed of <math>590</math> miles/hour. How many miles in two hours
-<math>\textbf{(A)}\ -3 \qquad\textbf{(B)}\ 0 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 7</math>
-==Problem 2==
-What is the numerical value of the sum <math>\sum_{k = 1}^{11}(i^{3} + i^{2})</math>
-<math>\textbf{(A)}\ 4000 \qquad\textbf{(B)}\ 4608 \qquad\textbf{(C)}\ 4862 \qquad\textbf{(D)}\ 5792 \qquad\textbf{(E)}\ 6100</math>
 ==Problem 3==
@@ Line 23: / Line 15: @@
 ==Problem 4==
-<math>10</math> cows can consume <math>40</math> kilograms of grass in <math>20</math> days. How many more cows are required such that all the cows together can consume <math>60</math> kilograms of grass in <math>10</math> days?
+A spaceship flies in space at a speed of <math>s</math> miles/hour and the spaceship is paid <math>d</math> dollars for each <math>100</math> miles traveled. It’s only expense is fuel in which it pays <math>\frac{d}{2}</math> dollars per gallon, while going at a rate of <math>h</math> hours per gallon. Traveling <math>3s</math> miles, how much money would the spaceship have gained?
 <math>\textbf{(A)}\ 20 \qquad\textbf{(B)}\ \ 21 \qquad\textbf{(C)}\ \ 22 \qquad\textbf{(D)}\ \ 23 \qquad\textbf{(E)}\ 24</math>
 ==Problem 5==
-Lambu the Lamb is tied to a post at the origin <math>(0, 0)</math> on the real <math>xy</math> plane with a rope that measures <math>6</math> units. <math>2</math> wolves are tied with ropes of length <math>6</math> as well, both of them being at points <math>(6, 6)</math>, and <math>(-6, -6)</math>. What is the area that the lamb can run around without being in the range of the wolves?
+Let <math>R(x)</math> be a function satisfying <math>R(m + n) = R(m)R(n)</math> for all real numbers <math>n</math> and <math>m</math>. Let <math>R(1) = \frac{1}{2}.</math> What is <math>R(1) + R(2) + R(3) + … + R(1000)</math>?
-<math>\textbf{(A)}\ 70 \qquad\textbf{(B)}\ 71 \qquad\textbf{(C)}\ 72 \qquad\textbf{(D)}\ 100 \qquad\textbf{(E)}\ 110</math>
 ==Problem 6==
@@ Line 45: / Line 31: @@
 ==Problem 7==
-Let <math>T(n)</math> denote the sum of the factors of a positive integer <math>n</math>. What is the sum of the <math>3</math> least possible values of <math>x</math> such that <math>T(x) + T(2x) = 8</math>?
+A Regular Octagon has an area of <math>18 + 18\sqrt {2}</math>. What is the sum of the lengths of the diagonals of the octagon?
+==Problem 8==
-==Problem 8==
+What is the value of <math>sin(1^\circ)sin(3^\circ)sin(5^\circ)…sin(179^\circ) - sin(181^\circ)sin(182^\circ)…sin(359^\circ)</math>?
+==Problem 9==
+Let <math>E(x)</math> denote the sum of the even digits of a positive integer and let <math>O(x)</math> denote the sum of the odd digits of a positive integer. For some positive integer <math>N</math>, <math>3E(3N)</math> = <math>4O(4N)</math>. What is the product of the digits of the least possible such <math>N</math>?
-The real value of <math>n</math> that satisfies the equation <math>ln(n) + ln(n^{2} - 34) = ln(72)</math> can be written in the form <cmath>a + \sqrt{b}</cmath> where <math>a</math> and <math>b</math> are integers. What is <math>a + b</math>?
+==Problem 10==
+In how many ways can <math>n</math> candy canes and <math>n + 1</math> lollipops be split between <math>n - 4</math> children if each child must receive atleast <math>1</math> candy but no child receives both types?
-<math>\textbf{(A)}\ 10 \qquad\textbf{(B)}\ 16 \qquad\textbf{(C)}\ 18 \qquad\textbf{(D)}\ 21 \qquad\textbf{(E)}\ 24</math>
+==Problem 11==
+Let <math>ABCD</math> be an isosceles trapezoid with <math>\overline{AB}</math> being parallel to <math>\overline{CD}</math> and <math>\overline{AB} = 5</math>, <math>\overline{CD} = 15</math>, and <math>\angle ADC = 60^\circ</math>. If <math>E</math> is the intersection of <math>\overline{AC}</math> and <math>\overline{BD}</math>, and <math>\omega</math> is the circumcenter of <math>\bigtriangleup ABC</math>, what is the length of <math>\overline{E\omega}</math>? Source: JHMMC 2019
-==Problem 9==
-Let <math>R(x)</math> denote the number of trailing <math>0</math>s in the numerical value of the expression <math>x!</math>, for example, <math>R(5) = 1</math>   since <math>5! = 120</math> which has <math>1</math> trailing zero. What is the sum
+<math>\textbf{(A)} \frac {31}{12}\sqrt{2} \qquad \textbf{(B)} \frac {35}{12}\sqrt{3} \qquad \textbf{(C)} \frac {37}{12}\sqrt{5} \qquad  \textbf{(D)} \frac {39}{12}\sqrt{7} \qquad \textbf{(E)} \frac {41}{12}\sqrt{11} \qquad  </math>
-<math>R(20) + R(19) + R(18) + R(17) + … + R(3) + R(2) + R(1) + R(0)</math>?
+==Problem 12==
+For some positive integer <math>k</math>, let <math>k</math> satisfy the equation
-<math>\mathrm{(A) \ } 38 \qquad \mathrm{(B) \ } 42\qquad \mathrm{(C) \ } 46\qquad \mathrm{(D) \ } 50\qquad \mathrm{(E) \ } 54</math>
+<math>log(k - 2)! + log(k - 1)! + 2 = 2 log(k!)</math>.
+What is the sum of the digits of <math>k</math>?
+==Problem 13==
-==Problem 10==
+An alien walks horizontally on the real number line starting at the origin. On each move, the alien can walk <math>1</math> or <math>2</math> numbers the right or left of it. What is the expected distance from the alien to the origin after <math>10</math> moves?
-In how many ways can <math>10</math> candy canes and <math>9</math> lollipops be split between <math>8</math> children if each child must receive atleast <math>1</math> candy but no child receives both types?
+==Problem 14==
+Let <math>K</math> be the set of solutions to the equation <math>(x + i)^{10} = 1</math> on the complex plane, where <math>i = \sqrt -1</math>. <math>2</math> points from <math>K</math> are chosen, such that a circle <math>\Omega</math> passes through both points. What is the least possible area of <math>\Omega</math>?
-==Problem 11==
+==Problem 15==
-Let <math>ABCD</math> be an isosceles trapezoid with <math>\overline{AB}</math> being parallel to <math>\overline{CD}</math> and <math>\overline{AB} = 5</math>, <math>\overline{CD} = 15</math>, and <math>\angle ADC = 60^\circ</math>. If <math>E</math> is the intersection of <math>\overline{AC}</math> and <math>\overline{BD}</math>, and <math>\omega</math> is the circumcenter of <math>\bigtriangleup ABC</math>, what is the length of <math>\overline{E\omega}</math>?
+Let <math>N = 10^{10^{100…^{10000…(100  zeroes)}}}</math>. What is the remainder when <math>N</math> is divided by <math>629</math>?
-<math>\textbf{(A)} \frac {31}{12}\sqrt{3} \qquad \textbf{(B)} \frac {35}{12}\sqrt{3} \qquad \textbf{(C)} \frac {37}{12}\sqrt{3} \qquad  \textbf{(D)} \frac {39}{12}\sqrt{3} \qquad \textbf{(E)} \frac {41}{12}\sqrt{3} \qquad  </math>
+==Problem 16==
-==Problem 12==
+Let <math>V</math> and <math>F</math> be the vertex and focus of the Parabola <math>P(x) = \frac{1}{8} x</math> respectively. For a point <math>G</math> lying on the directrix of <math>P(x)</math>, and a point <math>H</math> lying on <math>P(x)</math>, <math>\overline {GH} = 10</math> and Quadrilateral <math>VFGH</math> is cyclic. If <math>VFGH</math> has integral side lengths, what is the minimum possible area of <math>VFGH</math>?
-Rajbhog, Aditya, and Suman are racing a <math>1000</math> meter race. Aditya beats Rajbhog by <math>9</math> seconds and beats Suman by <math>250</math> meters. Given that Rajbhog beat Suman by <math>2</math> seconds, by how many meters would Aditya beat Rajbhog if they both were having a <math>3500</math> meter race?
+==Problem 17==
+Let <math>H(n)</math> denote the <math>2nd</math> nonzero digit from the right in the base - <math>10</math> expansion of <math>(2n + 1)!</math>, for example, <math>H(2) = 1</math>. What is the sum of the digits of <math>\prod_{k = 1}^{2020}H(k)</math>?
+==Problem 18==
-==Problem 13==
+<math>\bigtriangleup ABC</math> lays flat on the ground and has side lengths <math>\overline{AB} = 3, \overline{BC} = 4</math>, and <math>\overline{AC} = 5</math>. Vertex <math>A</math> is then lifted up creating an elevation angle with the triangle and the ground of <math>60^{\circ}</math>. A wooden pole is dropped from <math>A</math> perpendicular to the ground, making an altitude of a <math>3</math> Dimensional figure. Ropes are connected from the foot of the pole, <math>D</math>, to form <math>2</math> other segments, <math>\overline{BD}</math> and <math>\overline{CD}</math>. What is the volume of <math>ABCD</math>?
-In how many ways can the first <math>15</math> positive integers; <math>\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15\}</math> in red, blue, and green colors if no <math>3</math> numbers <math>a, b</math>, and <math>c</math> are the same color with <math>a + b - c</math> being even?
+<math>\textbf{(A) } 180\sqrt{3} \qquad \textbf{(B) } 15 + 180\sqrt{3} \qquad \textbf{(C) } 20 + 180\sqrt{5} \qquad \textbf{(D) } 28 + 180\sqrt{5} \qquad \textbf{(E) } 440\sqrt{2}</math>
-==Problem 14==
+==Problem 19==
-A function <math>f:\mathbb{C}\to\mathbb{C}</math> (Function along Complex Numbers) is defined by <math>f(z) = (a + bi)z</math>, where <math>a</math> and <math>b</math> are positive real numbers. The function <math>f(z)</math> has the property that there are complex numbers <math>z</math> such that <math>z, f(z)</math>, and <math>f^{2}(z)</math> form a triangle with an area of <math>20</math>. Given that <math>a + b = 10</math> and <math>(ab)^{2} = 30</math>, there is a certain complex number <math>c + di</math> where <math>d</math> is minimized as much as possible satisfying this condition. What is <math>c + d</math>?
+Let <math>P(x)</math> be a cubic polynomial with integral coefficients and roots <math>\cos \frac{\pi}{13}</math>, <math>\cos \frac{5\pi}{13}</math>, and <math>\cos \frac{7\pi}{13}</math>. What is the least possible sum of the coefficients of <math>P(x)</math>?
+==Problem 20==
-==Problem 15==
+What is the maximum value of  <math>\sum_{k = 1}^{6}(2^{x} + 3^{x})</math>   as <math>x</math> varies through all real numbers to the nearest integer?
-Let <math>N = 10^{10^{100…^{10000…(100  zeroes)}}}</math>. What is the remainder when <math>N</math> is divided by <math>629</math>?
+<math>\textbf{(A)}\ -3\qquad\textbf{(B)}\ -2\qquad\textbf{(C)}\ -1\qquad\textbf{(D)}\ 0\qquad\textbf{(E)}\ 1</math>
+==Problem 21==
-==Problem 16==
+Let <math>\lfloor x \rfloor</math> denote the greatest integer less than or equal to <math>x</math>. How many positive integers <math>x < 2020</math>, satisfy the equation
+ <math>\frac{x^{4} + 2020}{108} = \lfloor \sqrt (x^{2} - x)\rfloor</math>?
-For some positive integer <math>k</math>, let <math>k</math> satisfy the equation
-<math>log(k - 2)! + log(k - 1)! + 2 = 2 log(k!)</math>.
+==Problem 22==
-What is the sum of the digits of <math>k</math>?
+A convex hexagon <math>ABCDEF</math> is inscribed in a circle. <math>\overline {AB}</math> <math>=</math> <math>\overline {BC}</math> <math>=</math> <math>\overline {AD}</math> <math>=</math> <math>2</math>. <math>\overline {DE}</math> <math>=</math> <math>\overline {CF}</math> <math>=</math> <math>\overline {EF}</math> <math>=</math> <math>4</math>. The measure of <math>\overline {DC}</math> can be written as <math>\frac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. What is <math>m + n</math>?
-<math>\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 14 \qquad\textbf{(E)}\ 19</math>
+==Problem 23==
-==Problem 17==
+==Problem 24==
-How many ordered pairs <math>(a, b)</math> satisfy
+A sequence <math>(a_n)</math> is defined as <math>a_1 = \frac{1}{\sqrt{3}}</math>, <math>a_2 = \sqrt{3}</math>, and for all <math>n > 1</math>,
-<math>a\cdot b + lcm(a, b) = 12\cdot gcd(a, b) - a\cdot b</math>
+<math>a_{n + 1} = \frac{2a_n-1}{1 - a_n^2}</math>
-where <math>gcd</math> denotes the greatest common divisor and <math>lcm</math> denotes the least common multiple?
+What is <math>\lfloor \ a_{2020}\rfloor</math>?
-==Problem 18==
-<math>\bigtriangleup ABC</math> lays flat on the ground and has side lengths <math>\overline{AB} = 8, \overline{BC} = 15</math>, and <math>\overline{AC} = 17</math>. Vertex <math>A</math> is then lifted up creating an elevation angle with the triangle and the ground of <math>60^{\circ}</math>. A wooden pole is dropped from <math>A</math> perpendicular to the ground, making an altitude of a <math>3</math> Dimensional figure. Ropes are connected from the foot of the pole, <math>D</math>, to form <math>2</math> other segments, <math>\overline{BD}</math> and <math>\overline{CD}</math>. What is the volume of <math>ABCD</math>?
 ==Problem 25==
-Let <math>\lfloor x \rfloor</math> denote the greatest integer less than or equal to <math>x</math>. How many real numbers <math>x</math> satisfy the equation <math>x^4 + 20736\lfloor x \rfloor = 20736x</math>?
+Let <math>P(x) = x^{2020} + 2x^{2019} + 3x^{2018} + … + 2019x^{2} + 2020x + 2021</math> and let <math>Q(x) = x^{4} + 2x^{3} + 3x^{2} + 4x + 5</math>. Let <math>U</math> be the sum of the <math>kth</math> power of the roots of <math>P(Q(x))</math>. It is given that the least positive integer <math>y</math>, such that <math>3^{y} > U</math> is <math>2021</math>. What is <math>k</math>?

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